Figure 8

Shaped-pulse filter constructions. Construction of the first-order Walsh amplitude modulated dephasing-suppressing filter using shaped pulse segments. (a), (c), (e) Schematic representation of Walsh synthesis for a four segment gate of discrete Gaussian or trapezoidal segments. Walsh synthesis determines the overall amplitude of individual pulses with fixed duration and standard deviation, setting the effective pulse area in each segment. The metric g takes value \(1/6\) in panel (a), and \(1/12\) in panel (c). (e) Trapezoidal pulses are characterized by a constant slope such that all angles are a fraction of a square waveform defined as \(F\frac{\pi}{2}\). Here \(F=0.992\). In all pulse constructions the pulse profile is computed over 100 discrete time steps, permitting calculation of relevant filter-transfer functions. (b), (d), (f) Two-dimensional representation of the integral metric defining our target cost function, \(A(\boldsymbol {\Gamma }_{4})\) integrated over the stopband \(\omega\in[10^{-9}, 10^{-6}]\tau^{-1}\) for the corresponding pulse forms above. Areas in blue minimize \(A(\boldsymbol {\Gamma }_{4})\), representing effective filter constructions. The \(X_{0}\) determines the net rotation enacted in a gate while \(X_{3}\) determines the modulation depth, as represented in (a).